Real Numbers Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Classify each number as rational or irrational, and state whether it is a real number: −7-7, 9\sqrt{9}, 7\sqrt{7}, π\pi.

Solution

  1. 1
    −7=−71-7 = \frac{-7}{1} is rational. 9=3=31\sqrt{9} = 3 = \frac{3}{1} is rational. Both are real.
  2. 2
    7\sqrt{7} is irrational (7 is not a perfect square); π\pi is irrational (proven). Both are real.
  3. 3
    All four are real numbers: R=Q∪(R∖Q)\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q}).

Answer

All four are real numbers; −7, 9 rational; 7, π irrational\text{All four are real numbers; } {-7,\,\sqrt{9}} \text{ rational; } {\sqrt{7},\,\pi} \text{ irrational}
Every rational and every irrational number is a real number. The real numbers form the complete number line with no gaps. Being 'real' does not mean 'rational' — most real numbers are in fact irrational.

About Real Numbers

The complete set of all rational and irrational numbers, filling every point on the continuous number line.

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More Real Numbers Examples

Example 2 medium

Show that between any two distinct real numbers [formula] and [formula] (with [formula]), there exis

Example 3 easy

Which of the following are NOT real numbers? [formula], [formula], [formula], [formula].

Example 4 medium

Give one example of a real number between [formula] and [formula].

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